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Magazine Chapter 4: Problem 11
In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{r} x+3 y=4 \\ 4 x+5 y=2 \end{array}\right.$$
Short Answer
Expert verified
The solution in set notation is \{(-2,2)\}
Step by step solution
01
Prepare the equations
Before using the addition method, firstly rewrite both equations so they are in the form Ax + By = C. Given equations in the exercise already satisfy this condition: \[x+3y = 4\] and \[4x+5y = 2\]. There is no need for rearrangement this time.
02
Multiply the equations for elimination
Now, to initiate the addition method, multiply each equation by a suitable number so when they are added, one of the variables will cancel out. Here, choose to eliminate 'x' by multiplying the first equation by 4 and the second by -1 to get oppositely equal coefficients for 'x'. This results in: \[4(x+3y) = 4*4\] and \[-1(4x+5y) = -1*2\]. Therefore, the new equations become: \[4x+12y = 16\] and \[-4x-5y = -2\].
03
Add the equations
Now, add these two equations together such that the 'x' terms cancel each other out: \( (4x + 12y) + (-4x - 5y) = 16 + -2 \). This simplifies to: \(7y = 14\).
04
Solve for y
After simplification, solve the final equation, \(7y = 14\), for 'y' by dividing both sides by 7, yielding \( y = 2 \).
05
Substitute y into the original equation
Next, substitute the newfound 'y' value into the original equation \(x+3y =4\), yielding \( x + 3*2 = 4\). This simplifies to \( x + 6 = 4 \).
06
Solve for x
Solve the new equation, \(x+6 = 4\), for 'x' by subtracting 6 from both sides, resulting in \( x = -2\).
07
Format the solution in set notation
Finally, represent the solution in set notation; here, the solution set for this linear system is \( \{(-2,2)\} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Addition Method
The addition method is a popular strategy for solving systems of linear equations. It's also commonly known as the elimination method. The fundamental idea is to add two equations together in such a way that one of the variables gets eliminated. This simplifies the system and helps find the values of the remaining variables.
To use the addition method effectively, you should:
To use the addition method effectively, you should:
- Arrange the equations into a standard format (Ax + By = C).
- Determine which variable to eliminate.
- Multiply each equation by appropriate numbers so that when added, one variable cancels out.
- Add the equations directly to solve for the remaining variable.
Solution Sets
In mathematics, a solution set is a group of potential solutions that satisfy a system of equations. Each element of the set is a point that makes all equations in the system true.
For a system of linear equations, like the one you're solving, the solution set typically consists of:
For a system of linear equations, like the one you're solving, the solution set typically consists of:
- A single point \( (x, y) \) if there's a unique solution.
- No points if there is no possible solution (system is inconsistent).
- A set of infinitely many points if there are infinite solutions (dependent system).
Set Notation
Set notation is a mathematical way of representing a collection of objects. In the context of systems of equations, set notation is used to neatly express the solution set. It's a concise way to show answers and is widely accepted in mathematics.
For example, if a system of equations has a single solution, it is represented in set notation as \((x, y)\). If there are multiple solutions, those might be represented as a larger set of points. Using our specific example from the exercise, the solution had a single point: \
For example, if a system of equations has a single solution, it is represented in set notation as \((x, y)\). If there are multiple solutions, those might be represented as a larger set of points. Using our specific example from the exercise, the solution had a single point: \
- The solution set was \({(-2, 2)}\), indicating that x = -2 and y = 2 solve both equations.
Elimination Method
The elimination method is another name for the addition method and is a crucial concept in solving systems of equations. The main aim is to "eliminate" one of the variables to make it easier to solve for the others.
The process generally involves:
The process generally involves:
- Choosing a variable to eliminate based on simplicity or preference.
- Multiplying the equations by necessary constants to equalize coefficients of the chosen variable in opposing values.
- Adding or subtracting the equations to cancel out the chosen variable, simplifying the system into a single variable equation.
